When I worked on staff at associations, I was quite good at navigating organizational politics, particularly later in my career. What I realized was that organizational politics is simply a means to an end rather than the point of work. Once I figured that out, it was quite liberating and actually allowed me to engage in less politics while being more effective.
A critical part of managing organizational politics is understanding whether you are arguing about goals or about methods. Make this part of the conversation. Do we agree on what we are trying to achieve? If not, resolve that before moving on to determining how you will achieve it.
I have personally seen groups arguing about methodology when they had no agrement on what they were attempting to achieve. Going back to resolve the goal would often evaporate the original conflict because it was suddenly and obviously irrelevant to the clarified goal.
If you don’t clarify the core of the argument like this the only way to ‘win’ is by retaining your turf. This rarely creates value for the organization nor your mission.